Title | Solutions Manual Discrete-Event System Simulation Fourth Edition |
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Solutions Manual Discrete-Event System Simulation Fourth Edition Jerry Banks John S. Carson II Barry L. Nelson David M. Nicol January 4, 2005 Contents 1 Introduction to Simulation 1 2 Simulation Examples 5 3 General Principles 19 4 Simulation Software 20 5 Statistical Models in Simulation 21 6 Queue...
Solutions Manual Discrete-Event System Simulation Fourth Edition Jerry Banks John S. Carson II Barry L. Nelson David M. Nicol January 4, 2005
Contents 1 Introduction to Simulation
1
2 Simulation Examples
5
3 General Principles
19
4 Simulation Software
20
5 Statistical Models in Simulation
21
6 Queueing Models
36
7 Random-Number Generation
44
8 Random-Variate Generation
49
9 Input Modeling
54
10 Verification and Validation of Simulation Models
60
11 Output Analysis for a Single Model
62
12 Comparison and Evaluation of Alternative System Designs
66
13 Simulation of Manufacturing and Material Handling Systems
71
14 Simulation of Computer Systems
72
1
Foreword There are approximately three hundred exercises for solution in the text. These exercises emphasize principles of discrete-event simulation and provide practice in utilizing concepts found in the text. Answers provided here are selective, in that not every problem in every chapter is solved. Answers in some instances are suggestive rather than complete. These two caveats hold particularly in chapters where building of computer simulation models is required. The solutions manual will give the instructor a basis for assisting the student and judging the student’s progress. Some instructors may interpret an exercise differently than we do, or utilize an alternate solution method; they are at liberty to do so. We have provided solutions that our students have found to be understandable. When computer solutions are provided they will be found on the text web site, www.bcnn.net, rather than here. Instructors are encouraged to submit solutions to the web site as well. Jerry Banks John S. Carson II Barry L. Nelson David M. Nicol
Chapter 1
Introduction to Simulation For additional solutions check the course web site at www.bcnn.net. 1.1
a.
SYSTEM Small appliance repair shop
ENTITIES Appliances
ATTRIBUTES Type of appliance
ACTIVITIES Repairing the appliance
EVENTS Arrival of a job
STATE VARIABLES Number of appliances waiting to be repaired
Completion of a job Arrival at service line
Status of repair person busy or idle Number of diners in waiting line
Departures from service line Arrival at checkout counters
Number of servers working
Age of appliance
b.
c.
d.
Cafeteria
Grocery store
Laundromat
Diners
Shoppers
Washing machine
Nature of problem Size of appetite
Selecting food
Entree preference
Paying for food
Length of grocery list
Checking out
Breakdown rate
Repairing a machine
Departure from checkout counter Occurrence of breakdowns Completion of service
1
Number of shoppers in line Number of checkout lanes in operation
Number of machines running Number of machines in repair Number of Machines waiting for repair
2
CHAPTER 1. INTRODUCTION TO SIMULATION
e.
f.
g.
SYSTEM Fast food restaurant
ENTITIES Customers
Hospital emergency room
Taxicab company
Patients
Fares
ATTRIBUTES Size of order desired
Attention level required
Origination
ACTIVITIES Placing the order
EVENTS Arrival at the counter
STATE VARIABLES Number of customers waiting
Paying for the order Providing service required
Completion of purchase Arrival of the patient
Number of positions operating Number of patients waiting
Departure of the patient Pick-up of fare
Number of physicians working Number of busy taxi cabs
Traveling
Destination
h.
Automobile assembly line
Robot welders
Speed
Spot welding
Drop-off of fare Breaking down
Number of fares waiting to be picked up Availability of machines
Breakdown rate
1.3 Abbreviated solution: Iteration
Problem Formulation
1
Cars arriving at the intersection are controlled by a traffic light. The cars may go straight, turn left, or turn right.
2
Same as 1 above plus the following: Right on red is allowed after full stop provided no pedestrians are crossing and no vehicle is approaching the intersection.
3
Same as 2 above plus the following: Trucks arrive at the intersection. Vehicles break down in the intersection making one lane impassable. Accidents occur blocking traffic for varying amounts of time.
Setting of Objectives and Overall Project Plan How should the traffic light be sequenced? Criterion for evaluating effectiveness: average delay time of cars. Resources required: 2 people for 5 days for data collection, 1 person for 2 days for data analysis, 1 person for 3 days for model building, 1 person for 2 days for running the model, 1 person for 3 days for implementation. How should the traffic light be sequenced? Criterion for evaluating effectiveness: average delay time of cars. Resources required: 2 people for 8 days for data collection, 1 person for 3 days for data analysis, 1 person for 4 days for model building, 1 person for 2 days for running the model, 1 person for 3 days for implementation. How should the traffic light be sequenced? Should the road be widened to 4 lanes? Method of evaluating effectiveness: average delay time of all vehicles. Resources required: 2 people for 10 days for data collection, 1 person for 5 days for data analysis, 1 person for 5 days for model building, 1 person for 3 days for running the model, 1 person for 4 days for implementation.
CHAPTER 1. INTRODUCTION TO SIMULATION 1.4
3
Data Collection (step 4) - Storage of raw data in a file would allow rapid accessibility and a large memory at a very low cost. The data could be easily augmented as it is being collected. Analysis of the data could also be performed using currently available software. Model Translation (step 5) - Many simulation languages are now available (see Chapter 4). Validation (step 7) - Validation is partially a statistical exercise. Statistical packages are available for this purpose. Experimental Design (step 3) - Same response as for step 7. Production Runs (step 9) - See discussion of step 5 above. Documentation and Reporting (step 11) - Software is available for documentation assistance and for report preparation.
1.5 Data Needed Number of guests attending Time required for boiling water Time required to cook pasta Time required to dice onions, bell peppers, mushrooms Time required to saute onions, bell peppers, mushrooms, ground beef Time required to add necessary condiments and spices Time required to add tomato sauce, tomatoes, tomato paste Time required to simmer sauce Time required to set the table Time required to drain pasta Time required to dish out the pasta and sauce Events Begin cooking ¾ Complete pasta cooking Simultaneous Complete sauce cooking
Arrival of dinner guests Begin eating Activities Boiling the water Cooking the pasta Cooking sauce Serving the guests State variables
Number of dinner guests Status of the water (boiling or not boiling) Status of the pasta (done or not done) Status of the sauce (done or not done)
CHAPTER 1. INTRODUCTION TO SIMULATION 1.6 Event Deposit Withdrawal Activities Writing a check Cashing a check Making a deposit Verifying the account balance Reconciling the checkbook with the bank statement
4
Chapter 2
Simulation Examples For additional solutions check the course web site at www.bcnn.net. 2.1 Clock
Clock
Clock Time Waiting
Interarrival
Customer
Service
Time
Customer
Time
Time
Spends in
Idle Time
Time
Arrival
Time
Service
in Queue
Service
System
of Server
(Minutes)
Time
(Minutes)
Begins
(Minutes)
Ends
(Minutes)
(Minutes)
0
25
0
0
25
25
0
50
25
25
75
75
1 2
0
0
3
60
60
37
75
15
112
52
0
4
60
120
45
120
0
165
45
8
5
120
240
50
240
0
290
50
75
6
0
240
62
290
50
352
112
0
7
60
300
43
352
52
395
95
0
8
120
420
48
420
0
468
48
25
9
0
420
52
468
48
519
99
0
10
120
540
38
540
0
578
38
21
Average
45
19
112
(a) The average time in the queue for the 10 new jobs is 19 minutes. (b) The average processing time of the 10 new jobs is 45 minutes. (c) The maximum time in the system for the 10 new jobs is 112 minutes. 2.2 Profit = Revenue from retail sales - Cost of bagels made + Revenue from grocery store sales - Lost profit. Let Q = number of dozens baked/day X S= 0i , where 0i = Order quantity in dozens for the ith customer i
Q − S = grocery store sales in dozens, Q > S
S − Q = dozens of excess demand, S > Q
5
6
CHAPTER 2. SIMULATION EXAMPLES Profit = $5.40 min(S, Q) − $3.80Q + $2.70(Q − S) − $1.60(S − Q) Number of Customers 8 10 12 14
Dozens Ordered 1 2 3 4
Probability .35 .30 .25 .10
Probability .4 .3 .2 .1
Cumulative Probability .35 .65 .90 1.00
Cumulative Probability .4 .7 .9 1.0
RD Assignment 01-35 36-65 66-90 91-100
RD Assignment 1-4 5-7 8-9 0
Pre-analysis E(Number of Customers) = .35(8) + .30(10) + .25(12) + .10(14) = 10.20 E(Dozens ordered)
=
.4(1) + .3(2) + .2(3) + .1(4) = 2
E(Dozens sold)
=
S¯ = (10.20)(2) = 20.4
¯ Q) − $3.80Q + $2.70(Q − S) ¯ − $1.60(S¯ − Q) E(Profit) = $5.40Min(S, = $5.40Min(20.4, Q) − $3.80Q + $2.70(Q − 20.4) −$0.67(20.4 − Q) E(Profit|Q = 0)
= 0 − 0 + $1.60(20.4) = −$32.64
E(Profit|Q = 10) = $5.40(10) − $3.80(10) + 0 − $1.60(20.4 − 10) = −$0.64 E(Profit|Q = 20) = $5.40(20) − $3.80(20) + 0 − $1.60(20.4 − 20) = $15.36 E(Profit|Q = 30) = $5.40(20.4) − $3.80(30) + $2.70(30 − 20.4) − 0 = $22.08 E(Profit|Q = 40) = $5.40(20.4) − $3.80(40) + $2.70(40 − 20.4) − 0 = $11.08
The pre-analysis, based on expectation only, indicates that simulation of the policies Q = 20, 30, and 40 should be sufficient to determine the policy. The simulation should begin with Q = 30, then proceed to Q = 40, then, most likely to Q = 20.
7
CHAPTER 2. SIMULATION EXAMPLES
Initially, conduct a simulation for Q = 20, 30 and 40. If the profit is maximized when Q = 30, it will become the policy recommendation. The problem requests that the simulation for each policy should run for 5 days. This is a very short run length to make a policy decision. Q = 30 Day
RD for Customer
Number of Customers
RD for Demand
Dozens Ordered
1
44
10
8 2 4 8 1 6 3 0 2 0
3 1 1 3 1 2 1 4 1 4
Revenue from Retail $ 16.20 5.40 5.40 16.20 5.40 10.80 5.40 21.60 5.40 21.60
21
113.40
Lost Profit $ 0 0 0 0 0 0 0 0 0 0 0
For Day 1, Profit = $113.40 − $152.00 + $24.30 − 0 = $14.30
Days 2, 3, 4 and 5 are now analyzed and the five day total profit is determined.
2.3 For a queueing system with i channels, first rank all the servers by their processing rate. Let (1) denote the fastest server, (2) the second fastest server, and so on. Arrival event
No Server (1) busy?
Served by (1)
Yes No Served by (2)
Server (2) busy?
Yes No Served by (i)
Yes Server (i) busy?
Unit enters queue for services
Departure event from server j
Begin server j idle time
No
Another unit waiting?
Yes
Remove the waiting unit from the queue
Server j begin serving the unit
8
CHAPTER 2. SIMULATION EXAMPLES 2.4 Time Between Calls 15 20 25 30 35
Service Time 5 15 25 35 45
Probability .14 .22 .43 .17 .04
Probability
Cumulative Probability .12 .47 .90 .96 1.00
.12 .35 .43 .06 .04
First, simulate for one taxi for 5 days. Then, simulate for two taxis for 5 days.
¾
Cumulative Probability .14 .36 .79 .96 1.00
RD Assignment 01-14 15-36 37-79 80-96 97-00
RD Assignment 01-12 13-47 48-90 91-96 97-00
Shown on simulation tables
Comparison Smalltown Taxi would have to decide which is more important—paying for about 43 hours of idle time in a five day period with no customers having to wait, or paying for around 4 hours of idle time in a five day period, but having a probability of waiting equal to 0.59 with an average waiting time for those who wait of around 20 minutes. One Taxi Day
Call
Time between Calls 20 15 25 25 25
Call Time
1 2 3 4 5 6
RD for Time between Calls 15 01 14 65 73 48
1
Service Time
0 20 35 60 85 110
RD for Service Time 01 53 62 55 95 22
. . . 20
77
25
Time Customer Waits 0 0 20 20 20 30
Time Service Ends 5 55 80 105 140 155
Time Customer in System 5 25 45 45 55 45
Idle Time of Taxi
5 25 25 25 35 15
Time Service Begins 0 20 55 80 105 140
444
63
25
470
25
495
50
0
2 . . .
Typical results for a 5 day simulation: Total idle time = 265 minutes = 4.4 hours Average idle time per call = 2.7 minutes Proportion of idle time = .11 Total time customers wait = 1230 minutes Average waiting time per customer = 11.9 minutes Number of customers that wait = 61 (of 103 customers) Probability that a customer has to wait = .59 Average waiting time of customers that wait = 20.2 minutes
0 0 0 0 0 0
9
CHAPTER 2. SIMULATION EXAMPLES Two taxis (using common RDs for time between calls and service time)
Day
Call
Call Time
Service Time
1 2 3 4 5 6
Time between Calls 20 15 25 25 25
1
0 20 35 60 85 110
5 25 25 25 35 15
. . . 20
20
480
25
Taxi 1 Service Time
Time Service Begins 0 20
5 25
Time Service Ends 5 45
25 35
85 120
Time Service Begins
Taxi 2 Service Time
35 60 80
25
110
480
25
15
Time Service Ends
60
125
505
Time Customer Waits 0 0 0 0 0 0
Time Customer in System 5 25 25 25 35 15
Idle Time Taxi 1
0
25
10
2 . . .
Typical results for a 5 day simulation: Idle time of Taxi 1 = 685 minutes Idle time of Taxi 2 = 1915 minutes Total idle time = 2600 minutes = 43 hours Average idle time per call = 25.7 minutes Proportion of idle time = .54 Total time customers wait = 0 minutes Number of customers that wait = 0 2.5 X Y Z
= 100 + 10RN Nx = 300 + 15RN Ny = 40 + 8RN Nz
Typical results...
1 2 3 4 5 .. .
RN Nx -.137 .918 1.692 -.199 -.411
X 98.63 109.18 116.92 98.01 95.89
RN Ny .577 .303 -.383 1.033 .633
Y 308.7 304.55 294.26 315.50 309.50
RN Nz -.568 -.384 -.198 .031 .397
Z 35.46 36.93 38.42 40.25 43.18
W 11.49 11.20 10.70 10.27 9.39
2.6 Value of B
Probability
Cumulative
RD
Value of
Probability
Assignment
C
Probability
Cumulative
RD
Probability
Assignment
0
0.2
0.2
1-2
10
0.1
0.1
1-10
1
0.2
0.4
3-4
20
0.25
0.35
11-35
2
0.2
0.6
5-6
30
0.5
0.85
36-85
3
0.2
0.8
7-8
40
0.15
1
86-1
4
0.2
1
9-0
Idle Time Taxi 2
35 15 50
10
CHAPTER 2. SIMULATION EXAMPLES
Customer 1 2 3 4 5 6 7 8 9 10 Average
A
B
C
D
79.23 113.04 58.53 99.68 87.15 91.05 66.97 104.88 61.6 98.92 86.1
2 3 0 0 0 1 1 3 1 3 1.4
30 30 20 20 10 40 30 30 30 30 27
2 32 1.46 2.49 4.36 0.83 0.7 0.5 0.61 0.4 4.53
2.7 Lead Time (Days) 0 1 2 3 4 5
Probability .166 .166 .166 .166 .166 .166
Cumulative Probability .166 .332 .498 .664 .830 .996
RD Assignment 001-166 167-332 333-498 499-664 665-830 831-996 996-000 (discard)
Assume 5-day work weeks. D D
Week
Day
1
1 2 3 4 5 6 7 8 9 10
2
Beginning Inventory 18 15 11 7 2 0 0 13 11 7
= Demand = 5 + 1.5(RN N )( Rounded to nearest integer)
RN N for Demands -1.40 -.35 -.38 .05 .36 .00 -.83 -1.83 -.73 -.89
Demand 3 4 4 5 6 5 4 2 4 4
Ending Inventory 15 11 7 2 0 0 0 11 7 3
.. .
Typical results Average number of lost sales/week = 24/5 = 4.8 units/weeks
Order Quantity
RD for Lead Time
Lead Ti...